Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ceqsaltv Structured version   Visualization version   GIF version

Theorem bj-ceqsaltv 33001
Description: Version of bj-ceqsalt 33000 with a dv condition on 𝑥, 𝑉, removing dependency on df-sb 1938 and df-clab 2638. Prefer its use over bj-ceqsalt 33000 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ceqsaltv ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-ceqsaltv
StepHypRef Expression
1 bj-elissetv 32986 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
213anim3i 1269 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∃𝑥 𝑥 = 𝐴))
3 bj-ceqsalt0 32998 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
42, 3syl 17 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054  wal 1521   = wceq 1523  wex 1744  wnf 1748  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-ex 1745  df-nf 1750  df-clel 2647
This theorem is referenced by:  bj-ceqsalgvALT  33006
  Copyright terms: Public domain W3C validator